Lesson 4

Diego said that he thinks that \(\sqrt{3}\approx 2.5\).

1. Use the square to explain why 2.5 is not a very good approximation for \(\sqrt{3}\). Find a point on the number line that is closer to \(\sqrt{3}\). Draw a new square on the axes and use it to explain how you know the point you plotted is a good approximation for \(\sqrt{3}\).

2. Use the fact that \(\sqrt{3}\) is a solution to the equation \(x^2 = 3\) to find a decimal approximation of \(\sqrt{3}\) whose square is between 2.9 and 3.1.

The numbers \(x\), \(y\), and \(z\) are positive, and \(x^2 = 3\), \(y^2 = 16\), and \(z^2 = 30\).

1. Plot \(x\), \(y\), and \(z\) on the number line. Be prepared to share your reasoning with the class.
2. Plot \(\text- \sqrt{2}\) on the number line.

Here is a line segment on a grid. What is the length of this line segment?

By drawing some circles, we can tell that it’s longer than 2 units, but shorter than 3 units.

To find an exact value for the length of the segment, we can build a square on it, using the segment as one of the sides of the square.

The area of this square is 5 square units. (Can you see why?) That means the exact value of the length of its side is \(\sqrt5\) units.

Notice that 5 is greater than 4, but less than 9. That means that \(\sqrt5\) is greater than 2, but less than 3. This makes sense because we already saw that the length of the segment is in between 2 and 3.

In general, we can approximate the values of square roots by observing the whole numbers around it, and remembering the relationship between square roots and squares. Here are some examples:

• \(\sqrt{65}\) is a little more than 8, because \(\sqrt{65}\) is a little more than \(\sqrt{64}\) and \(\sqrt{64}=8\).
• \(\sqrt{80}\) is a little less than 9, because \(\sqrt{80}\) is a little less than \(\sqrt{81}\) and \(\sqrt{81}=9\).
• \(\sqrt{75}\) is between 8 and 9 (it’s 8 point something), because 75 is between 64 and 81.
• \(\sqrt{75}\) is approximately 8.67, because \(8.67^2=75.1689\).

If we want to find a square root between two whole numbers, we can work in the other direction. For example, since \(22^2 = 484\) and \(23^2 = 529\), then we know that \(\sqrt{500}\) (to pick one possibility) is between 22 and 23.

Many calculators have a square root command, which makes it simple to find an approximate value of a square root.